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Theorem eltrrels2 34873
 Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
eltrrels2 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))

Proof of Theorem eltrrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dftrrels2 34869 . 2 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
2 coideq 34564 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
3 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
42, 3sseq12d 3859 . 2 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
51, 4rabeqel 34573 1 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ⊆ wss 3798   ∘ ccom 5346   Rels crels 34526   TrRels ctrrels 34538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-rels 34783  df-ssr 34796  df-trs 34866  df-trrels 34867 This theorem is referenced by:  eltrrelsrel  34875
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